## 11 bilateral scientific research cooperation projects with the USA awarded to UP IAM and UP Famnit

**The Departments of Mathematics at UP IAM **and** UP Famnit **were once again successful in the call for proposals launched by the Slovenian Research Agency (ARRS) for co-financing scientific research cooperation between the Republic of Slovenia and the United States of America in the period 2022-2024.

In total, the University of Primorska has obtained the co-funding for 12 new projects, **11 of which are in the field of mathematics**. In addition, one more bilateral cooperation project has been awarded to Dr. Michael Mrissa, who works at the InnoRenew CoE and the University of Primorska. Dr. Tina Vukasović has obtained a project at the Faculty of Management.

The ARRS calls for proposals co-finance mutual visits between Slovenian and US researchers and researchers to facilitate joint scientific research projects aimed at strengthening the cooperation between the two countries and increasing the mobility of researchers. In addition, the aim is to increase the number of projects submitted by Slovenian researchers to EU Framework Programme calls as well as to other international calls for R&D programmes.

The following projects, led by UP IAM and UP Famnit researchers, will be completed by the end of June 2024.

Title: **Chemical graphs on steroids**

Institutions: UP IAM and The Massachusetts College of Liberal Arts (MCLA)

UP project leader: **Dr. Nino Bašić***Fullerenes and benzenoids are important chemical compounds. In mathematical chemistry, they are modelled by fullerene graphs and benzenoid graphs. Much of the mathematical chemistry literature is concerned with their mathematical properties. The research will proceed in several directions. In the present project we will refine the notion of the Pentagonal Incidence Partition. Both of the topics will also be considered in relation to Clar and Fries numbers, two graph invariants of fullerenes and benzenoids. We also plan to investigate the computational complexity of determining the Clar number (and several associated values) in fullerenes and nanotubes.*

Title: *New bounds in extremal graph theory *

Institutions: UP IAM and Mississippi State University

UP project leader: **Dr. Slobodan Filipovski***In this project we will establish new bounds on the maximum number of edges in K_r - free graphs in terms of the order n and the maximum degree d_1, in terms of the order n and the minimum degree d_n, and in terms of n and both d_1 and d_n. Moreover, we will show that the bounds obtained in this project are better than the corresponding bounds of Mantel and Turán. More precisely, we show that the upper bounds on the maximum sizes of K_r -free graphs of specified maximum or minimum degrees given by our theorems are smaller than or equal to the upper bounds on the maximum sizes among all K_r -free graphs. *

Title: *Distance-regular graphs with exactly three irreducible T-modules with endpoint 1, all** of which are thin*

Institutions: UP IAM and Seattle University

UP project leader: **Dr. Štefko Miklavič***Our research concerns a combinatorial object known as a graph. A graph is a finite set of vertices, together with a set of undirected arcs or edges, each of which connects a pair of distinct vertices. We say that vertices x, y are adjacent whenever x, y are connected by an edge. The concept of a graph is useful because mathematical, as well as intuitive notions, can be formulated in terms of adjacency. Our research concerns a type of graph said to be distance-regular.*

Title: *Studies in duality: graphs on surfaces and conformal hypergraphs*

Institutions: UP IAM and Rutgers University, MSIS Department, RUTCOR

UP project leader:** Dr. Martin Milanič***The project will focus on the study of selected open problems in graph and hypergraph theory. Graph theory is a young branch of discrete mathematics that has developed rapidly in recent decades, largely due to many applications in the modern world in such diverse fields as computer science, sociology, biology, logistics, and so on. A graph is a combinatorial object consisting of a finite set of vertices and a set of edges each connecting two vertices. A hypergraph is a generalization of a graph where an edge can be formed by any subset of vertices. The project will address two central research topics related to two types of duality.*

Title: *Combinatorial structure and algebraic properties of a Q-polynomial graph for which** the adjacency algebra is closed*

Institutions: UP IAM and University of Wisconsin

UP project leader: **Dr. Safet Penjić***The alternating central extension of q-Onsager is generated by q-Onsager along with some extra central elements. This raises a question that the project leader plan to work on. Our problem is: Can we find any elements in our algebra \T, that are not in T, but commute with everything in \T? Such elements might be the missing generators for the alternating central extension of Q-Onsager or U^+_q.*

Title: *Matroids with Transitive Automorphism Group*

Institutions: UP Famnit and University of Mississippi

UP project leader:** Dr. Edward Taucher Dobson***Questions regarding the symmetry of combinatorial objects have a long history and are still very active research topics. Usually, these questions are asked about digraphs and graphs, sometimes about designs such as Steiner triple systems, and occasionally about error-correcting codes. We propose to consider these, and related questions, for matroids, a combinatorial object which is a joint generalization of graphs and linear algebra. *

Title: *Certain structural properties of Cayley graphs*

Institutions: UP Famnit and Kennesaw State University

UP project leader: **Dr. Ademir Hujdurović***We plan to investigate strong independent sets and strong cliques in Cayley graphs. Cliques in Cayley graphs are also interesting from the point of view of studying the intersection density of permutation groups, namely intersecting sets of a permutation group G correspond to cliques of the Cayley graph on G with the connection set S being the union of all point stabilizers.*

Title: *Computer Algorithm Development for Molecular Dynamics Simulation of Macromolecules*

Institutions: UP Famnit and Kennesaw State University and National Institutes of Health (NIH)

UP project leader: **Dr. Dušanka Janežič***The goal of the proposed research is to introduce new improvements in molecular dynamics simulation of macromolecules that increase the accuracy and efficiency of present-day simulation approaches. Our goal is to develop new methodological solutions for prediction and study of protein binding sites, based on graph theoretical approaches, combined with molecular dynamics simulations. One of the goals of this project is to combine ProBiS web server with molecular dynamics methods into a new freely available web tool: ProBiS-ATLAS: Protein interaction atlas for prediction of genetic variations involved in drug interactions and disease development enabling the discovery of molecules, relevant to pharmaceutical research.*

Title: *Certain problems in hypergraphs, graphs, and games*

Institutions: UP Famnit and Rutgers University, MSIS Department, RUTCOR

UP project leader: **Dr. Matjaž Krnc***In the course of this project, we plan to focus on the following: Problem 1: The Bi-SP conjecture. Problem 2: On the avoidable paths. A path is called avoidable if every linear extension may be extended to an induced cycle. Problem 3: An efficient characterization of threshold 3-uniform hypergraphs. A hypergraph is said to be threshold if there exists a linear weight function on the vertices separating the independent sets from the dependent ones. Problem 4: Deterministic graphical games with no Nash equilibria. We plan to **analyze** the existence of deterministic graphical games that have no Nash equilibria in pure stationary strategies, with respect to additive effective payoffs. Problem 5: Exact transversals in graphs and hypergraphs. We plan to study the related exact transversal hypergraph operator, study conditions under which it is involutive, and examine connections with strong cliques in graphs and related concepts.*

Title: *Preserver problems in quantum mechanics*

Institutions: UP Famnitand the Department of Mathematics, Embry–Riddle Aeronautical University

UP project leader: **Dr. Bojan Kuzma***Spectrum preservers on unbounded operators. In a mathematical formulation, quantum-mechanical observables (i.e., things that can be observed and measured) are modelled with (unbounded) self-adjoint operators acting on a complex separable Hilbert space. As part of the project, we intend to classify all additive transformations that map (unbounded) self-adjoint operators into themselves and preserve their spectrum. Isometries of complexified norms. Any real vector space can be complexified, ie. made into a complex vector space by extending the scalars; the procedure is unique. In contrast to this, for a given norm on a real space, we can define several different complex norms on the complexified space, all of which extend the initial real norm. Preserver problems in quantum information sciences. Tensor products play a key role in quantum mechanics and quantum information theory: In mathematical terminology, a quantum state is described by a positive-semidefinite matrix of trace-one; the tensor product of two (or more) states describes a joint bipartite (or multipartite) system.*

Title: *Computational aspects of some combinatorial games on Ferrer's diagrams*

Institutions: UP Famnit and Rhodes College

UP project leader: **Dr. Riste Škrekovski***Combinatorial game theory is a large and growing field that includes in its scope a wide range of game types, generally focusing on two-player games in which both players have perfect information and there are no moves of chance. Sprague and Grundy introduced a method of quantifying game positions for impartial normal-play games, i.e., those in which both players have the same possible moves in each position. These Sprague-Grundy values are a generalization of winning and losing positions. *